In this paper we looked at strongly minimal linearizations for any given rational matrix \(R(\lambda )\) , preserving the structure whenever we have a specific type of self-conjugate structure on \(R(\lambda )\) . We showed that there always exist strongly minimal linearizations that have the same self-conjugate structure as \(R(\lambda )\) . These results were known for the case of proper rational matrices [1]}, [2]}, but were extended here to rational matrices that are not proper. Moreover, the derivation is new and is based on arguments that are very similar for the strictly proper part and the polynomial part of the rational matrix. The proofs are also constructive and lead to efficient algorithms for the construction of strongly minimal linearizations both in the unstructured and in the structured cases. The fact that the proposed construction of structure preserving strongly minimal linearizations is valid for any structured rational matrix improves significantly all the results available in the literature for constructing other classes of structured preserving linearizations of the structured rational and polynomial matrices considered in this paper, which are not valid for all rational or polynomial matrices (see, for instance, [3]}, [4]}, [5]}, [1]}, [7]}, [8]} and the references therein). Interesting future work motivated by the results in this paper may include the extension of the construction of structured strongly minimal linearizations to other classes of structured rational or polynomial matrices and to rational matrices whose polynomial part is expressed in bases different from the monomial basis.
